Choosing a Function Model

Every polynomial family leaves a fingerprint in evenly spaced data. Take first differences between consecutive outputs. If those are constant, the data is linear, degree 1. If not, take differences of the differences, the second differences. Constant second differences mean quadratic, degree 2. Constant third differences mean cubic, and so on. This works because the nth difference of a degree n polynomial strips away everything but the leading term's effect, leaving a constant tied to the leading coefficient.

This matters because a function model should capture the pattern driving the data, not just memorize the points you happened to measure. With only a handful of points, you can always find some high-degree polynomial that threads through every single one exactly. That does not mean the data came from a degree 6 process. Real measurements carry noise, and a model with too many degrees of freedom bends itself to fit that noise instead of the trend. A lower-degree model that comes close to most points, without hitting every one, is often the more honest description of what is really going on.

This is the payoff of the whole unit: once you know the degree, you know the end behavior, the possible number of turning points, and how the leading term will dominate far from the data. Choosing the model is choosing which of those behaviors you are betting on when you extrapolate beyond the table.